Monty Hall problem: probability puzzle, solution, and significance
A probability puzzle originating from the TV show Let's Make a Deal. Explains the three-door setup, why switching doubles the win probability, historical controversy, and common variants.
The Monty Hall problem is a counterintuitive probability puzzle named after the host of the American game show Let's Make a Deal. It poses a simple game: a contestant chooses one of three doors; behind one door is a valuable prize and behind the other two are lesser "goats" or booby prizes. The host, who knows what is behind every door, opens one of the remaining doors to reveal a goat, then offers the contestant the choice to stay with the original pick or switch to the other unopened door. Readers can find general background on probability at probability resources, the show at Let's Make a Deal, and the problem's namesake at Monty Hall.
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Problem statement
In precise steps the game proceeds as follows: the host hides a single high-value prize behind one door and two low-value prizes behind the others; the contestant selects a door but does not open it; the host—aware of all door contents—opens one of the two remaining doors and always reveals a goat; finally, the contestant is invited to either keep the original choice or switch to the only other unopened door. For clarity, the toy model often describes the valuable item as a car and the decoys as goats; see a discussion of typical prize framing at prize examples and the decoys at goat examples. The central question is: does switching doors change the contestant's chance of winning the car?
Why switching helps
Many people's first intuition is that after one goat is revealed the two remaining doors each have an equal 50/50 chance. That impression ignores the host's knowledge and deliberate choice. A simple way to see the correct answer is to list the equally likely initial choices and track outcomes if the contestant always switches:
- Contestant initially picks the door with the car (probability 1/3). Host opens a goat door. If the contestant switches, they move to a goat and lose.
- Contestant initially picks goat A (probability 1/3). Host must open the other goat (if rules force revealing a goat), leaving the car behind the remaining closed door. Switching yields the car and a win.
- Contestant initially picks goat B (probability 1/3). By symmetry, switching again yields the car and a win.
Because two of the three initial possibilities lead to a car when switching, the switch strategy wins with probability 2/3, while staying wins with probability 1/3. This result hinges on the host always opening a door that he knows contains a goat and never opening the contestant's chosen door.
Another way to understand it is to note that the contestant's first choice has a 1/3 chance of being correct and a 2/3 chance of being wrong. When the host opens a goat door, he transfers no new probability to the chosen door; instead, the unopened, unchosen door inherits the 2/3 chance that the contestant's original pick was wrong. Computer simulations and repeated trials quickly confirm that switching wins roughly twice as often as staying.
History, controversy and variants
The puzzle gained widespread public attention in the early 1990s after being discussed in public columns and academic writing. The published solution provoked strong disagreement from many readers who found the 2/3 result surprising; subsequent explanations, visualizations and simulations helped settle the dispute for most observers. Variants of the problem explore what happens when the host's behavior is altered—for example, if the host sometimes opens the car door by mistake, if the host chooses a door at random when two goats are available, or if there are many more than three doors. Each change requires restating the host's rules explicitly: the 2/3 conclusion depends on the host deliberately avoiding the car and always offering the switch.
The Monty Hall problem is useful as a teaching example because it highlights conditional probability, the importance of information and how an agent's knowledge and actions affect outcomes. It also serves as a caution about trusting immediate intuition: carefully accounting for the process that generated the revealed information often changes the probabilities in non‑obvious ways.
The experiential response
If you ask the question to people who had not yet dealt with the problem, they often assume that the chances of winning for gates 1 and 2 are the same. The reason often given for this is that nothing is known about the motivation of the showmaster to open gate 3 with a goat behind it and to offer a change. Therefore, the indifference principle applies.
Intuition in understanding the letter to the editor assumes that the problem statement is a description of a unique game situation. In addition, the response evidences some familiarity with game shows such as Go for it, in which the showmaster (host) plays an active and unpredictable role. In contrast to the problem variants in which the presenter is reduced to a 'stooge' bound by fixed rules of behaviour, it may be realistically assumed that he is completely free to make his own decisions (Monty Hall: "I'm the man of the house!"). This freedom can be illustrated by a few examples, where before each game the car and goats were randomly redistributed behind the three gates. Because the candidates are familiar with this game show, for which they have applied as participants, they are naturally aware of the unpredictability of the host.
Game 1
Candidate Alfred chooses gate 1, the moderator opens gate 1 with a goat behind it; Alfred loses.
Game 2
Candidate Berta chooses gate 1, the presenter opens gate 2 with a goat behind it and offers Berta to change her choice. Berta wants to change, but the moderator does not open a gate, but offers 5000 euros for Berta to stay with her first choice. Berta does not change her choice and the moderator opens gate 3 with a goat behind it; Berta loses.
Game 3
Candidate Conny chooses gate 1, the presenter does not open a gate, but offers the candidate 1000 Euros in exchange for not opening the gate; Conny takes the money and wins 1000 Euros.
Game 4
Candidate Doris chooses gate 1, the presenter then opens gate 3 with a goat behind it and offers Doris to reconsider her choice ...
Given the various ways the moderator can behave, Doris should carefully consider her chances of winning. If she thinks the moderator is being nice to her and wants to distract her from her first wrong choice, then she should switch. However, if she thinks the moderator is not being kind to her and just wants to distract her from her first correct choice, then she should stay with Gate 1. If Doris can't gauge the moderator - no such clues are given in the letter to the editor either - then she has no way of correctly calculating her chances of winning. In particular, once the moderator intervenes, she can no longer invoke the indifference principle, and so the answer to the question "Is there an advantage to changing your choice of gate?" in her case is "Not necessarily."
Although this already answers the letter to the editor's question, the suggestion was made to support Doris in her decision and give her a real 50:50 chance of winning. This is done by assuming that she has the option of choosing one of the two remaining goals after tossing a fair coin. In this way, she can ensure that her probability of winning is exactly 1⁄2 regardless of the moderator's intentions.
Reply from Marilyn vos Savant
Due to Marilyn vos Savant's response to the letter to the editor, the problem achieved a high level of attention internationally, even outside the professional community, and led to heated controversy. Her response was:
"Yes, you should switch. The first gate you choose has a 1⁄3 chance of winning, but the second gate has a 2⁄3 chance of winning. Here's a good way to visualize what happens. Suppose there were 1 million gates and you chose gate number 1. Then the presenter, knowing what is behind the gates and always avoiding the one gate with the prize, opens all the gates except gate number 777777. You would immediately switch to that gate, wouldn't you?"
Marilyn vos Savant does not take into account any particular motivation of the host; according to the letter to the editor, it is not impossible that the host opens a goal gate only to distract the candidate from his first, successful election. Instead, vos Savant apparently takes the letter to the editor to mean that the game show follows the same pattern over and over again:
Course of the game show: The respective candidate chooses a gate, the presenter then always opens another gate with a goat behind it and then gives the candidate another choice between the two gates that are still closed. The candidate receives the car if it is behind the gate he last chose.
Thus, it obtains as a solution the average probability of winning all possible combinations of gates chosen by each candidate, which can then be opened by the moderator. Because the first choice of a candidate is considered arbitrary and the distribution of car and goats behind the gates is considered random, each of the nine possibilities may be considered equally probable:
| Gate 1 selected | Gate 2 | Gate 3 | Moderator opens ... | Result when changing | Retention result |
| Car | Goat | Goat | Gate 2 or Gate 3 | Goat | Car |
| Goat | Car | Goat | Gate 3 | Car | Goat |
| Goat | Goat | Car | Gate 2 | Car | Goat |
| Gate 1 | Gate 2 selected | Gate 3 | |||
| Car | Goat | Goat | Gate 3 | Car | Goat |
| Goat | Car | Goat | Gate 1 or Gate 3 | Goat | Car |
| Goat | Goat | Car | Gate 1 | Car | Goat |
| Gate 1 | Gate 2 | Gate 3 selected | |||
| Car | Goat | Goat | Gate 2 | Car | Goat |
| Goat | Car | Goat | Gate 1 | Car | Goat |
| Goat | Goat | Car | Gate 1 or Gate 2 | Goat | Car |
Three out of nine candidates win by sticking with their first choice, while six out of nine candidates get the car by switching. Thus, by switching, a candidate has an average chance of winning of p = 2⁄3.
This solution can also be illustrated graphically. In the pictures of the following table the chosen gate is arbitrarily shown as the left gate:
| Behind the first selected gate is the car | Behind the first chosen gate is a goat | |
| Probability 1⁄3 | Probability 2⁄3 | |
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| The moderator opens one of the gates with a goat (doesn't matter which one!) | The moderator can only open the other gate with goat | |
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| Switching leads to the gain of a goat | Switching leads to winning the car | |
| Switching is disadvantageous with a probability of only 1⁄3 but advantageous in 2⁄3 | ||
Strategic solution
As a result, vos Savant's conception of gameplay can also be reproduced in the following way:
The respective candidate may determine two freely chosen gates, which the moderator must open, and receives the car if it is behind one of these two gates.
Procedure taking into account the switching strategy proposed by vos Savant:
For example, if a candidate wants to have gate 2 and gate 3 opened, he would first select gate 1, which remains locked. The candidate then moves to Gate 2 when the moderator has opened Gate 3, or vice versa. The candidate thus obviously has an average chance of winning of p = 2⁄3. Accordingly, it would always be advantageous for a candidate to switch gates.
Questions and answers
Q: What is the Monty Hall problem?
A: The Monty Hall problem is a famous probability (chance) problem based on a television game show from the United States, Let's Make a Deal. It involves three doors, one of which has a car behind it and two of which have goats behind them.
Q: What does the host know?
A: The host knows what is behind each door and always chooses to open a door with a goat behind it.
Q: Does changing choices increase the chances of getting the car?
A: Yes, changing choices increases the chances of getting the car from 1/3 (one out of three) to 2/3 (two out of three).
Q: How does this probability work?
A: In the initial door choice, there is only a 1/3 chance that the player will pick the door with the car. There is then a 2/3 chance that if they switch their choice after seeing one of the other doors opened by the host, they will get a car.
Q: Are all options equal in terms of winning or losing?
A: No, there are three different options for winning or losing depending on whether or not you change your choice after seeing one of other doors opened by the host. If you pick correctly initially and then switch your choice, you will lose; if you pick incorrectly initially but switch your choice afterwards, you will win; and if you pick correctly initially but don't switch your choice afterwards, you will also win.
Q: Is it true that switching increases your chances to win two times out of three?
A: Yes, it is true that switching increases your chances to win two times out of three.
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AlegsaOnline.com Monty Hall problem: probability puzzle, solution, and significance Leandro Alegsa
URL: https://en.alegsaonline.com/art/66510